Catastrophe Theory

By Salix alba – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=26446257

I’ve already covered the topic of fractals and Chaos Theory, but the arrival and popularity of these two obscures a slightly earlier and rather similar mathematical topic which has a number of things in common with them, although it’s a lot “smoother”. This was Catastrophe Theory.

On 28th July 1975, BBC-2 broadcast a ‘Horizon’ documentary entitled ‘Happy Catastrophe’ which got a larger response from its viewership than any other ‘Horizon’ episode. It clearly captured the public’s imagination, attracting more correspondence than any other ‘Horizon’ up until that point, and in fact stuck in my own mind more than most other programmes at the time. Looking back at it, I found a number of other episodes in the mid-’70s quite memorable, such as the one on epilepsy and another on Erich von Däniken, which I mention here, but certainly this is one of them, and in fact epileptic seizures themselves could be modelled using catastrophe theory (CT) itself. To an extent, I want to blog about CT today, but I’m also interested in why it was so popular, and why it seems to be largely ignored today.

CT deals with discontinuities, which are moments of sudden change. For example, if you take a thin card and press it at its sides, it will do very little for quite a while, then suddenly crumple or flip into a different shape, and letting go of the card will not lead to its return to anything like the flat form it had before, although it will tend to spring back a little. The same applies to a snapping rubber band under tension and a host of other situations, such as the epileptic seizures I mentioned just now, although one would hope in this last case that the brain can in fact fairly quickly return to a more organised state. Unfortunately this is rarely not so, in which case it becomes a medical emergency.

The programme’s title, ‘Happy Catastrophe’, is interesting. When we use the word in English, and it is of course a Greek word, we generally mean something negative. The Greek word, “καταστροφη”, consists of the words “κατα”, meaning “down”, and “στρεφειν” – to turn, in other words a “downturn”, and with the usual connotations of falling does indeed have negative connotations. The word was prominently used in drama, where it referred to the fourth and final part of a play, after protasis, epitasis and katastasis. We’re familiar with it today through tragedy, but in fact it also applied to comedy, and in that setting it referred to a happy ending such as a wedding. Hence our own usage has become predominantly negative, but for some time I attempted to use it with a more neutral connotation, which in fact makes the word a lot more useful, although it can be confusing and we don’t really have control over the meaning of words, particularly when we lack something like L’Académie Française. There were two types of catastrophe, whether happy or otherwise, in Greek drama. In a simple catastrophe, there’s simply a transition from dramatic events to a quieter set of circumstances without any change in character, unravelling or revelation. Complex catastrophes involve sudden discoveries by the character or sudden changes in fortune which are feasible and upon which the plot depends, rather than being a deus ex machina. In a way, simple catastrophes occupy one side of the graph whereas complex ones occupy the other. This is what I mean:

Taken from here. Will be removed on request.

A simple catastrophe can be thought of as a movement across the steady slope on the left hand side of this graph. It descends into repose without anything huge happening. I don’t know what examples there are of this but to be honest they sound a bit boring. Complex catastrophes, on the other hand, are movements along the right hand side of the graph and involve events “falling off a cliff” in such a way that they permanently change things. This graph is of course the “cusp catastrophe”. It makes me wonder what the variable labelled as “u” is in drama. ‘Œdipus Rex‘ definitely occupies the right hand side – it has a low value of u, whatever that might be. It’s also important to remember that if you turn this graph upside down, you more or less have the same graph, and that therefore comedies are also catastrophic in nature. ‘Much Ado About Nothing’ is just as catastrophic as ‘Œdipus Rex‘, but in a positive way.

Incidentally, in what I’ve just said I can’t help but be reminded of this:

Can you usefully take a quantitative approach to literature? In a way the answer is a definite “yes”, because for instance you could look at repetition of certain words and phrases or the prosody or rhyme scheme of a particular poem, but in general it does have a bad rap. But I can’t help noticing that when John Keating gets the pupils to rip out the introduction to ‘Understanding Poetry’, it is a catastrophic event, and of course later in the film there are other incidents more deserving of the word, but there’s no going back once the introduction has been ripped out, as the end of the film illustrates.

The cusp catastrophe graph looks like the kind of shape you’d get if you held a thin sheet of metal horizontally and bent it towards or away from you. This is because that situation is in fact a catastrophe with two control dimensions and one behaviour dimension. The buckling which occurs on one side of the sheet is dramatically greater than on the other. This now sounds like an engineering or metallurgy issue, but can be used for drama, as with the 1951 film ‘No Highway In The Sky’, which involves the catastrophic failure of aircraft in this way. In this case the behaviour axis involves the plane falling out of the sky and killing everyone, although there’s another catastrophe where Theodore Honey deliberately damages a plane to prevent it taking off and killing the occupants:

I’ve mentioned control and behaviour dimensions, or axes, without really explaining what they are. To elaborate, it makes sense to consider the simplest possible models, including non-catastrophic ones, which have two dimensions. A section of a two-dimensional line graph can have a number of shapes relevant to CT. It can be a slope, a trough, a peak or a fold. Except for the slope, these are all the same basic shape. With a fold, the shape is like a C rather than a U or an “n”. This means that as the control variable increases, the behaviour of the system can either become more dramatic or less so, to choose one possible label for a variable, but will be stuck in that trend unless the other variable reduces considerably. Or, it can be reflected along the Y axis and will be stuck in a trend unless that variable increases a lot. This is the “zone of inaccessibility” and can be shown in several other examples.

There are substances whose melting points are not the same as their freezing points. That is, if a solid of this nature is heated, it will melt at a particular point, but if the resultant liquid is then cooled, it may need to be made colder than the temperature at which it melted to solidify. I seem to remember that cocoa butter does this, but there are many examples. Similarly, when tuning in an analogue radio with a manual tuner, one can find a station, then tune up past it and then find that it seems to be on a lower frequency than one previously found it when twisting the knob back again. These are examples of the kind of behaviour which is modelled in the overhang found in the cusp catastrophe. A value can increase smoothly until it leaps to a higher value if another value is high, but can also stay on the lower surface, and likewise can stay on the higher surface until it is lower than when it initially leapt up. I have a feeling that tidiness is like this. It takes more effort to tidy something up in one big go than it appears to when one does it bit by bit, and then it slips down into untidiness more easily.

Adding a dimension clearly results in three-dimensional graphs, and again there are a certain number of these. Incidentally, before I go on I want to point out that CT graphs only focus on a narrow range of variables where something interesting is occurring, and are therefore small portions of potentially infinite graphs. The two-dimensional “fold” catastrophe could easily diverge to an ever-increasing but smooth extent along its control axis, even to infinity. Also, in illustrating these graphs the section can be a small map of a much larger landscape, such as one including peaks and basins or mountains and valleys. It’s just that the distinctive shapes can be broken down in this way.

Three-dimensional graphs could just be extensions of two-dimensional ones, so for example a valley could just be long and not do much interesting in the Z-axis, so all the types still exist in three and more dimensions and are not cancelled out by the new ones, but each added dimension does introduce additional graphs. In the three-dimensional case, X and Z can be the controls and Y the behaviour, which makes the surfaces more relatable as they’re more like topographical features. There’s the slope which rises diagonally to the axes, the peak, what I’m going to call the “crater”, which is a dent in a surface, and two less familiar shapes, the col and the cusp. I want to mention the col even though it isn’t catastrophic, because it’s less well-known or easy to relate to than the others.

A col is a gap between two peaks. These are often nameless locations, although passes are cols. They occur also in air pressure patterns, where there’s a low-pressure point between two high pressure weather systems. There’s also the saddle:

Saddles differ from cols in continuing to curve away in both directions, concave on one side and convex on the other. A col is the central point of a saddle according to one definition.

The cusp is crucially different from all of these because it has a kind of asymmetry to it along one axis, although it also is rotationally symmetrical in that turning it 180° around the axis labelled u in the earlier graph, assuming it’s aligned correctly, will lead the same shape. This mixture of asymmetry and symmetry doesn’t apply to the other shapes and the cusp is the only discontinuous shape involved.

These shapes appeal to the eye, and it’s been said that CT is particularly visual. It shares this feature with many fractals and the Mandelbrot Set, and in this respect serves as a kind of herald to those later, particularly visually appealing, mathematical excursions. It also has a kind of universalising tendency, which despite its name has been described as modelling rather than a theory. Calling it a theory is a bit like counting two legs on a person and seeing that there are two stars in a binary star system and calling that “integer theory”. It’s more that this kind of model can be applied to natural phenomena, and as seen above with the illustration of catastrophes as a dramatic device, also in the social sciences and humanities. The issue of their beauty may be similar to the beauty of regular fractals and the Mandelbrot Set, in that certain features echo the characteristics of being a product of the Universe, which is who we are in one respect.

There are a total of seven graphs, according to CT, which can between them be used to model all discontinuities. These are: the fold, cusp, butterfly, swallowtail, hyperbolic umbilic, parabolic umbilic and elliptic umbilic. The hyperbolic umbilic is illustrated at the start of this post, where it comprises the upper part of the image. Because it’s a five-dimensional shape, the illustration isn’t exactly what it “looks” like, but is in fact what’s known as the bifurcation set of the hyperbolic umbilic. This is a projection of the shapes which are discontinuous in the graph. In the case of the cusp, this is a kind of curved V-shape extending to infinity or the edge of the graph, like a kind of shadow cast by illumination on a transparent model, or alternatively, and this is more important than it might seem, the kind of light reflected by illuminating a smooth metallic version. The bifurcation set of an hyperbolic umbilic is like two superimposed half-pipes at a shallow angle to each other semicircular in cross-section at opposite ends smoothly becoming curved V-shapes at the other. That probably isn’t very clear. It has two behaviour dimensions rather than one, and three control dimensions. Umbilics are points on locally spherical surfaces, and hyperbolic ones have just one ridge line passing through the point in question, which if I’ve described the above clearly means the point of intersection between the two half-pipes. It’s interesting to contemplate what it would be like to skateboard around the bifurcation set of an hyperbolic umbilic.

The other two umbilics are the parabolic and elliptic. Elliptic umbilics have three control and two behaviour dimensions and the bifurcation set looks like a cross-sectionally curved triangular prism pinched smoothly to a point at the centre, which is the three ridge points passing through the umbilic point. Finally, the parabolic umbilic is six-dimensional, with four control and two behavioural dimensions, making it particularly hard to visualise as even the bifurcation set has four dimensions, but are transitional between hyperbolic and elliptic umbilics, with two ridges, one of which is singular. Visualised using the fourth dimension as time, running in one direction the bifurcation of a parabolic umbilic looks like a shrinking paper plane crashing through the fold in a sheet of paper folded into a V-shape while another V-folded paper shape at the bottom is flattening out and bowing outward.

The other two are the rather less awkwardly-named butterfly and swallowtail. The former is interestingly named because of the butterfly effect, but is not more closely linked to that than the others. It’s five-dimensional, with four control dimensions and one behaviour dimension, and has been used to model eating disorders. It looks odd, even reduced to three dimensions, which effectively destroys its usefulness but enables one to work out what it’s doing, as it looks like a cusp catastrophe with three cusps linked in a kind of triangle. That is, a triangle can be drawn between the three points where the cusps split off from the smooth side, but that triangle isn’t oriented in three-dimensional space unless the butterfly is rotated in such a way that most of it is in hyperspace.

The swallowtail catastrophe is so named because a mathematician was trying to describe it to a blind person, who responded that it sounded like a swallowtail, which it does. It’s merely four-dimensional and its bifurcation set looks like a swallowtail at one end with a U-shape above it with the tail diminishing into the U halfway along. This has one behaviour dimension and three of control. Salvador Dalí’s last painting, if it was his, in 1983, was based on this graph, and was entitled “The Swallow’s Tail”:

This is a cross-section of the bifurcation set with some extra bits added. The monoline S shape is a cross-section of the cusp catastrophe. Dalí described CT as “the most beautiful æsthetic theory in the world”. The artist used to kind of “riff” on scientific theories in an artistic way, using them as inspiration without necessarily understanding them in an analytical way. He also included a formula describing the swallowtail in his 1983 painting linked here entitled ‘El rapte topològic d’Europa. Homenatge a René Thom’. The last few years of his life are controversial because it’s alleged that he was made to sign canvases by his carers which would later be used to paint forgeries, and the above painting may not be his because his hands were said to be too shaky for him to draw such a line, which brings Britney Spears to my mind. After completing this painting, if he did, Dalí tried to enter a state of suspended animation through fasting and died five years later, soon after giving the visiting Juan Carlos a drawing entitled ‘The Head Of Europa’.

One way of looking at these graphs is to see the compartments as representing different stable states. Hence the six “cells” of the parabolic umbilic plus the seventh open space nearby are each conditions some systems can enter if there are four main factors determining their behaviour, which can in turn be described in terms of two factors. The same can be applied to the others.

I mentioned Dalí’s tremor making his creation of ‘The Swallowtail’ questionable, but in fact tremor and noise are not likely to disturb the behaviour of catastrophes. They’re quite stable in this respect, which calls into question the often-quoted explanation as to why they’re now so seldom modelled in this way being that not many systems can be adequately described with so few variables. This property is accompanied by what are called “attractors”, which CT has in common with Chaos Theory. An attractor is a set of states a system tends to drift towards, or in this case jump towards. Each one of the cells I mentioned just now is an attractor. After having got there, the system will tend to continue to be at least somewhat like that. It occurs to me in fact that limerence could be modelled in this way. It’s easy to get fixated on someone but it can be a lot harder to get over them. That, then, would be literally an attractor: a person one finds attractive. This suggests it would be fruitful to work out which control variables are involved, since in certain crucial circumstances, people do end up suffering from long-term limerence. However, discussing it and other psychological models in this way raises the question of positivism, which can be criticised on the grounds of reductivism.

You may or may not have heard of Gartree Prison, which was well-known for its helicopter escape in 1987. I have two personal connections to Gartree. One is that it ended up housing the bloke who abducted me in 1989 and the other is that one of my tutors on the herbalism course was married to a Gartree prison guard. Rather startlingly, Gartree prison disturbances were modelled using CT, more specifically the cusp catastrophe. This makes for a significant case study of the application of CT to social phenomena. When this was done, CT was riding on a wave of popularity triggered by the ‘Horizon’ broadcast and was possibly quite immature in its development, although as a modelling method it dates back to Edwardian times, the modelling having been published in 1976. The control variables seem to have been tension and alienation, which were assessed quantitatively, an approach which seems quite vague. They were based on governor applications, inmates requesting segregation, staff absenteeism, welfare visits and inmates in the punishment cells, and the shape of the graph seems to have been derived using a method which, it’s said, could have been made to fit almost any data set. There may have been an issue in the dominant connotations of the word “catastrophe” here, because it tends to be interpreted as negative and would perhaps consequently tend to lead to applications of the theory to model negatively-perceived events such as prison riots. It might also have been used by the prison service to make its operations and management appear more scientific than it actually was. And in any case, scientific management is widely regarded as a bad thing, at least for workers, as it’s seen as leading to redundancy, monotonous work, exploitation of workers, and from the management side expensive to implement, time-consuming and leading to a deterioration in quality. This could have implications for the situation inside prisons, as they are also workplaces for the staff and sometimes also for prisoners, so simply making the measures required might impair the function of the institution.

This could be applied more widely to other institutions such as mental hospitals and schools. For instance, if it successfully predicted grades in a school and also ways of manipulating variables in order to get those grades onto a higher tier of the graph, it wouldn’t necessarily improve less quantifiable measures of school performance. Likewise, a similar approach might lead to higher “cure” rates in a mental hospital, but that would only be in terms of particular paradigms of “abnormal” behaviour. Could it be applied to increase the quality of poetry? Maybe it could. Maybe J Evans-Pritchard would be able to measure the greatness of the poetry output by all these “cured” psychotics and high-achieving school-leavers with his scale. Or, maybe we just like to imagine that we aren’t reducible in such a way to a few variables and graphs, but maybe we’re wrong about that.

The modelling here, and in the other two as far as I know fictional examples I gave (the mental health one is less fictional than one might think), is applied to systems which depend on many assumptions about how society should be. For instance, assuming the prison study was valid, it might still fail to show anything because prisons of that kind are constrained by social factors always to be on one side of the cusp, and whereas manipulating the variables beyond that range is theoretically possible, doing so would not be possible given factors like level of public funding, policy regarding responses to crime and the nature of the buildings used. Then again, maybe we do want an entirely evidence-based set of policies. I would personally prefer that. It’s called socialism.

In a realm entirely outside the question of social policy, meditation, states of consciousness or mental illness, catastrophe graphs turn up in another rather surprising place: caustics. Caustics are projections of light rays reflected or refracted by a reflective or transparent medium onto a surface. I mentioned previously that a model of a cusp catastrophe could be made of mirror-like reflective material and be illuminated, and such a situation could lead to the projection of a caustic onto a flat screen. Caustics are the kind of light pattern you see when you look down into a clean, empty mug into which sunlight is shining, and they alter their shape and size according to the angle of incidence. They can also be seen in the dappling effect on a sandy seabed of waves on a sunny day. They can also have a kind of three-dimensional appearance, and in the teacup case they seem to look rather like a swallowtail bifurcation set, but in three dimensions in each case. Moving the cup leads to a different section of the graph. Caustics are odd because they’re always sharp and it isn’t clear what’s so special about the area they illuminate as opposed to its surroundings. They’ve also historically been problematic in computer graphics because depicting them accurately is computationally intensive, so in CGI they tend to be more decorative than realistic. It would be interesting to know whether catastrophe theory could simplify or has ever been used to generate caustics in computer images. Moreover, it would also be interesting to know if images of three-dimensional slices of higher-dimensional CT graphs could be accurately generated using three-dimensional reflective surfaces to generate their caustics.

A major question remains. Why don’t we hear so much about CT nowadays when it was so popular forty-odd years ago? An answer might be found in an illustration from herbalism, and at this point I shall intrepidly venture onto the territory of one of my other blogs. It’s been noted that herbal prescriptions with an odd number of remedies tend to be more successful than those with an even number. This needs to be restrained in various ways. For instance, it doesn’t mean that an even-numbered ℞ can be made more effective by omitting one of the herbs or adding one which is not relevant to the patient’s needs. I hypothesised that the reason for this was that an odd-numbered prescription could be modelled in terms of relative doses using catastrophe theory, whereas an even-numbered ℞ couldn’t. However, there are a number of problems with this which can be extended to other situations. The herbs here are presumed to be the control dimensions of the graph. A fold catastrophe has one control dimension, a cusp two, a swallowtail three, a butterfly four, a hyperbolic umbilic three, parabolic four and elliptic three, so the number of remedies would seem to have to be three or one if this is to hold true. In fact ℞s tend to have five or seven remedies, if one is in the low number of remedies in high doses as am I, because I feel the high number of remedies in low doses is beginning to look like homeopathy. Hence it can’t be applied to most herbal prescriptions other than simples, and there would have to be something which makes the fold, swallowtail and hyperbolic and elliptic umbilics distinctive in terms of their efficacy, which may be true but I’m not sure about that. But there’s a bigger problem which applies more widely. Herbs are not single remedies. They generally include a large number of different compounds with various effects on each other and physiology. Thus it seems implausible to apply catastrophe theory to herbalism, and this can be broadened out into biology more generally, since in most biological situations the number of control dimensions would be too high for CT to be relevant.

CT is still applicable to engineering and physics, but its intended target, the inexact sciences such as sociology, psychology and ecology, is rather more slippery. It does still happen, for instance in modelling the population dynamics of aphids via the butterfly catastrophe (it would have to be named after an insect – presumably the swallowtail is useful for modelling bird migration), but there really do seem to be too many variables and the smoothing effect initially claimed doesn’t seem to hold. That said, the formulæ used to generate the graphs are quite simple, and this could lend them to use in computer games, both in generating caustics on the graphics side and the likes of political and social interactions in games like Sim City.

The Expanding Earth Theory

Some time ago, I had a Peters Projection map of the world from ‘New Internationalist’ magazine. The idea behind the Gall-Peters Projection (it wasn’t originally Peters) is that it’s supposed to show all the land in correct proportion as to its size, which I presume it does, but the problem is that it doesn’t preserve compass direction and considerably distorts the shapes. As far as I can work out, it’s a cylindrical projection that differs from Mercator by progressively reducing the north-south lengths to zero as they approach the poles. There is a lot to be written about this map projection in particular and political correctness, but not here.

No, where I’m going with this is the reaction one of my friends had to the maps of continental drift in the book. There were a couple of pages showing the evolution of this planet’s appearance from the supercontinent Pangæa in the late Permian to its current appearance. Pangæa looked roughly like this:

There are certain issues with this map, such as the fact that Antarctica is ice-covered in it, which it wasn’t at the time, but it succeeds in roughly illustrating the supercontinent and the condition of the surrounding tectonic plates at that time. Again, Pangæa and the general idea of supercontinents is interesting but still not quite what I’m going to talk about today.

My friend made a rather disparaging comment regarding the map of Pangæa along the lines of it just being guesswork and “how the hell could they know this is what happened?” I explained to him about the continents fitting together, the presence of symmetrical bands of magnetic minerals on the floor of the Atlantic, the continuation of coal seams across continents which match the jigsaw and the presence of fossils of the same species in widely separated parts of the world. Incidentally, today I might add that Earth’s interior has now been found to be cooling faster on one side than the other, indicating that something large was blocking the heat for a long time in the distant past, and this is thought to be the above supercontinent. His response, after I’d said all this, was “yeah, but how the hell could they know this is what happened?”! It was like I hadn’t said anything!

To some extent, I think his attitude is a healthy one, and I presume it was based in distrust for authority of any kind. He’s an intelligent, well-educated guy and I’m not disparaging him for his opinion. It’s just that I feel that it illustrates something which I doubtless also do, where I reject counter-intuitive and novel ideas, sometimes just because they’re new. It’s a widespread phenomenon for people to receive a new idea, perhaps not listen very closely to the evidence cited in support of it and proceed to pick holes in it and reject it out of hand. This is all the more so when immediate observation seems to contradict it, as can be seen today with Flat Earthers. They have an approach they describe as “Zetetic Cosmology”, which is the idea that one should always depend on what can be directly observed oneself, and in many ways this is commendable, and like my friend involves distrust of authority, which is again to be encouraged. However, there comes a point when one either has to trust experts in a field other than one’s own or find an example of something which would prove one’s assumption wrong if it turned out to be true when tested. In the case of the Flat Earth, my answer is to use railway timetables in distant parts of the world and online traffic cams to observe daylight, because in both cases these being fake would involve a ridiculously vast conspiracy, lots of people missing important appointments and a whole load of RTCs. Other examples of this would include the idea that the Apollo missions were a hoax and the various Covid-19 conspiracy theories.

At the same time, it’s uncomfortable to have one’s world view challenged on the other side. I don’t know how far back my acceptance of continental drift goes, but I remember mentioning a piece of evidence for it in school in 1976, which is currently forty-five years ago when I was nine, so it’s one of those things which forms a kind of cherished part of the jigsaw I use to make sense of life, and it’s disturbing to have that questioned. Consequently, although I’m aware of lots of evidence supporting it, I probably use that evidence more as a comfort blanket to confirm that my beliefs about the world are correct rather than actually enquiring into it in any great depth. That does also mean I trust experts in this area. But there’s a psychological urge to force people into believing what I believe which is more about competition and perhaps aggression than altruism, and that’s not a good motive.

Alfred Wegener was an early proponent of the theory of continental drift. He noted that South America and Afrika seemed to fit into each other neatly, with Brazil jutting out in a shape very close to how the Gulf of Guinea “juts in”, and the Great Australian Bight matching the coastline of Victoria Land in Antarctica. He thought of this in around 1911.

Prior to this, and in fact for many decades after, the prevailing wisdom was that land bridges rose and sank between the different continents, causing flora and fauna to mix, which is for example why the continent of Lemuria in the Indian Ocean was posited. There are prosimians (non-simian primates such as tarsiers and bush babies) in Madagascar, continental Afrika and Indonesia, so how did they get to be in such widely separated places? The answer was supposed to be Lemuria, named after lemurs. Oddly, although this idea has now been discarded, there was in fact formerly a fairly large landmass in the Indian Ocean and in a few million years time there will be again, when the Afrikan Rift Valley opens up and East Afrika splits off. The descriptions of changes in geography in Olaf Stapledon’s ‘Last And First Men’ also relate land bridges rising and falling, as was generally believed in the 1930s. I even have a book from the late 1940s with a map of them as they were supposed to be in the Mesozoic, shown above.

Land bridges don’t really work though, because they violate the principle that crustal rocks generally float at the same level above the mantle depending on its depth and density. For land bridges to appear and disappear in that way, their density or thickness would have to change.

The problem with continental drift was that there didn’t seem to be a mechanism for them to move around. Wegener proposed something called Polflucht – “pole flight”. His idea was that the centrifugal effect of Earth’s rotation pulls the land masses away from the poles and causes them to break apart as they approach the equator. If this idea worked, it would make sense to a certain extent because we’re in a situation where the Tethys, an ocean which used to stretch all the way round the equator, has now closed due to the collision of Afrika and Eurasia and the formation of the Isthmus of Panama, and Australia has also moved north from its prior connection with Antarctica. The problem is, however, that the crust is far too sturdy to allow this to happen. It’s also interesting that Wegener, who was mainly an expert on polar geology, would focus on this aspect of the planet to explain.

At this time, as far as I can tell, there wasn’t any idea of a supercontinent cycle, where continents collide together and are broken up, only to join together again in a different configuration hundreds of millions of years later. The reason I say this is that the explanation which was proposed after this was rejected seems to suppose that Pangæa was the one and only original supercontinent which then broke up and the continents formed then drifted into their current positions. The idea proposed was of course the Expanding Earth Theory:


Author
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The idea here is that all the land was joined together when Earth was first formed, and this planet was considerably smaller back then. Then Earth expanded and the single landmass cracked apart, creating today’s world map. There were various hypotheses about how this might have happened, one of which I find a lot more interesting than the others. One is that Earth started off as the rocky core of a gas giant like Jupiter and was therefore compressed and under a lot of pressure. The Sun gradually boiled off the atmosphere and as the pressure reduced, the planet “sprang out” and expanded due to its release. Another theory is based on the idea of the luminiferous æther, which in itself probably could do with an explanation. It used to be thought that just as sound or waves in water need a medium to carry them, so did light, radio waves and the like, and this was referred to as the æther. Although this idea is not completely dead for complicated reasons which slip my mind right now, the æther’s existence was refuted by the Michelson-Morley experiment, which showed that light travels at the same speed whether or not it’s moving in the direction of our planet’s orbit or at right angles to it, meaning that there was no static medium carrying it and ultimately ushering in Einsteins theory of relativity. Incidentally this experiment is also used by Flat Earthers to “prove” that Earth does not orbit the Sun or rotate. Isaac Newton believed that gravity was caused by a condensation of the æther combined with its rarefaction, which was eventually applied to the idea of the atmosphere doing the same thing, thereby providing the basic theory for powered heavier-than-air flight by explaining lift. Æther was later demonstrated to be necessarily incompressible and it was thought that matter was a sink in this æther, an idea which was clearly on its way to becoming Einsteins theory of general relativity and in fact something I used to believe myself up until I was about thirteen. This was then elaborated by Ivan Yarkovsky into the suggestion that Earth gradually accumulated matter from the transmutation of the æther into atomic matter and therfore slowly expanded.

Those are the less interesting explanations. The one which I feel drawn to, although it isn’t true, is Dirac’s. Paul Dirac was one of the most important physicists of the last century and is extremely respected. He proposed that the gravitational constant was slowly decreasing, causing the planet to expand gradually. Once again this explains continental drift, and seems to develop fairly naturally out of Newton’s and Yarkovsky’s theories of gravitation, but it also does something else which is very interesting. It amounts to an explanation for the expansion of space, and therefore is quite economical and elegant in its explanatory power. It isn’t just about Earth but the whole cosmos.

There is an odd parallel between the Expanding Earth Theory and theories of the evolution of the Universe. Over the past century there have basically been four of these. One of the best supported is now refuted, which is the Steady State Theory. This is the idea that space is infinite and constantly expanding, with matter being generated slowly within it, so that at any one time the visible part of the Universe looks roughly the same. In this view, there was no beginning to the Universe and it will always exist. The established and widely accepted theory today is more or less the Big Bang Theory, which is that the Universe expanded out from a single point around 13 800 million years ago and will continue to expand forever. I have my issues with this but I won’t mention them today. If there was only slightly more mass than there in fact seems to be in the Universe, it would also end up collapsing into a similar state to the early Universe in the distant future. Finally, there is the “oscillating Universe”. This involves an endless series of collapses and expansions, and raises the philosophical question of whether time is cyclical or each instance of the Universe is a new one. Although the Big Bang Theory is the only really acceptable one among scientists at the moment, there is also a theory that the Big Bang was preceded by a collapsing Universe made of antimatter when time was running backwards, which sounds pretty similar to the oscillating Universe to me.

Just as there was an oscillating Universe theory, later discredited, there was also an oscillating Earth theory. This involved the planet going through alternating phases of expansion and contraction which explained the phenomena on this planet which look like they’re caused by contraction. I imagine this includes mountain ranges but that’s just my guess. I find it interesting that there were two cyclical expansion-contraction theories about the world, one involving Earth and the other the Universe.

It is of course very appealing that there should be a single explanation combining continental drift and the Big Bang Theory based on weakening gravity. I don’t know if this has ever been done, but it also strikes me as a good explanation for the fact that fossils of extinct life forms tend to be much bigger than the life forms around today, such as dinosaurs and giant insects. Maybe this is because the fossils themselves have expanded over time and back in the day, the animals and plants who became them were of relatively modest size. However, this is not so because the Expanding Earth Theory is refuted, and in science you have to be brutal about your emotional attachments. Dirac’s idea is absolutely lovely, but it’s also dead wrong.

I mentioned train timetables earlier as a way to refute the Flat Earth hypothesis. This works because a sphere cannot be mapped onto a flat surface without distortion, as illustrated by Peters Projection. This means that two distant train routes of the same length would in some cases be distorted on a map. The Flat Earth is effectively a map of the real Earth, because it’s a curved surface forced into a flattened shape. This means that somewhere on this flat Earth, notably in Canada and Australia according to the main idea Flat Earthers have Earth’s shape, it ought to take a lot longer to go the distance the route is supposed to cover than it actually does. Now it could simply be that the map shown above is wrong, but there will always be routes whose length is dramatically distorted if Earth has a continuous flat surface and Euclidean geometry is roughly applicable, because every map distorts the planet’s surface. This is a particularly reliable reason for saying Earth cannot be flat.

As it happens, the same kind of idea can be applied to the Expanding Earth theory. I mentioned previously that there are stripes of magnetic minerals on the floor of the Atlantic. These are generated when the ocean floor spreads out from the central ridge, which is volcanic. As magnetic materials float in the lava, they get lined up with Earth’s magnetic field, which varies in its direction and strength. These then solidify with their alignments pointing in particular directions, and they line up symmetrically because the ocean is spreading from a ridge running roughly down the middle in both directions. If Earth was expanding, these magnetic materials would line up as if they’re on a smaller planet the older they are, meaning that it would be like attempting to project a globe onto a larger one without changing the sizes of the map. They would not line up according to longitude.

Satellites are now able to measure the size of the planet to within two hundred microns and there is no expansion faster than that. Continental drift is faster than that at about an inch a year in some places, so the idea that Earth is expanding is redundant, as it fails to explain what’s going on. The continents are also moving in different directions. For instance, the Pacific is gradually narrowing, as is the Mediterranean, so there isn’t a general trend towards expansion.

The trouble with this evidence is that it starts to become a little abstract and therefore lays itself open to being distrusted. As soon as it becomes difficult to follow a line of argument, or where it involves trusting an expert in a different discipline from one’s own experience, the possibility of error or perhaps conspiracy arises. This isn’t necessarily something to be distrusted, but at the same time questioning and distrust is important. The ultimate solution may be to become as well-informed as possible on certain matters, and perhaps to be self-aware when one is overly attached to a particular view, and maybe question one’s motives. Because whatever else is true, Dirac’s version of the expanding Earth and its link with an expanding Universe is truly appealing, but it’s still turned out to be wrong. But it’s tough to accept this.

Beware The Nice Ones

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This may sound arrogant, but I have a reputation for being a “nice” person. Because that sounds like a positive thing, it could come across as narcissistic to say it, but in fact the reason I’m saying it now is that in fact, being “nice”, and I will rename that trait in a moment so you won’t have to deal with the incessant quote marks, is not always a good thing, not only for the person but also for those around them. There are also plenty of people who definitely don’t think I’m nice, so it’s okay.

“Niceness” is known in academic psychology as “agreeableness”, and is part of Five Factor Personality Theory. And at this point I will permit myself a foray elsewhere, into personality theories in general, because they occupy a peculiar epistemological position. But I’ll get to that.

Five Factor Theory is one of several theories of personality prevalent in academic psychology. The five factors are agreeableness, openness, extraversion, neuroticism and conscientiousness. It’s possible to remember these using the acronym OCEAN, in a different order. There is also the question of why they would be oriented in this direction, e.g. why not “closedness”, “introversion” and so forth? Even so, for the sake of argument it’s worthwhile to think of them a particular way round without considering the implications. Of the five, openness has been criticised as being rather meaningless, which it doesn’t seem to be to me.

Openness involves curiosity, the appreciation of beauty and the willingness to try new things. It contrasts with the opposite pole of dogmatism, lack of interest in new experiences and a data-driven approach. Neuroticism is a tendency to experience negative emotions and emotional instability. This in particular strikes me as having a bearing on the concept of personality disorder, and I’ll go into that here too. Extraversion is gaining energy from external means, breadth of experience, perceiving interaction with people positively, high visibility, talkativeness and so forth. Conscientiousness is diligence, tendency to focus on attempting to control one’s impulses and concern for one’s obligations. It’s important to note at this point that conscientiousness is not agreeableness, which is concern for social harmony, kindness, honesty, generosity and so forth – sympathy and niceness, basically.

Although I will get onto problematising agreeableness, I want to point out a couple of odd things about the status of personality theories. Some of them do fall under the umbrella of academic psychology and others do not. I’m going to ignore Freud here because his ideas are florid and not evidence based, although they are important from a cultural perspective and not entirely invalid, but there are others, notably Cattell’s sixteen-trait theory, which is important to me because it formed part of the inspiration for my first degree dissertation. Cattell gathered evidence from various sources and reduced the traits via a process known as factor analysis, which is a statistical method for reducing the number of variables, which, and this is important, tend to make them harder to grasp. I won’t go down that rabbit hole again right now and probably shouldn’t have then either. Outside the realm of academic psychology are Myers-Briggs and the Enneagram. In case you’re wondering, I’m supposed to be INTJ in Myers-Briggs and a four with a five wing in the Enneagram system. Myers-Briggs is reproducible, that is, repeated tests at considerable intervals produce the same results. I can more or less confirm this from my personal experience, because I took the predecessor of the MBTI test in 1979 and again sometime in the ‘noughties and came up with the same results, although the former test didn’t include all of the factors. There’s also reproducibility for me with the Enneagram, and since there’s also correlation between MBTI and the Enneagram, it can be expected that it’s also valid even though the two have radically different ethoi. My impression with the Enneagram, though, is that it tends to work better with more markèd personality traits. This too I will come back to.

It seems to me there’s an issue with some personality theories being inside the tent and others being outside it. It’s possible to look at Western astrology and the Far Eastern blood group personality hypothesis as personality theories as well, and of course both of those are generally rejected by sceptics and skeptics alike. The issue with astrology, which I’ve researched and found wanting incidentally, is generally said to be that there’s a kind of general, vague description of each personality type which focusses in its popular version mainly on the Sun Sign, which can be swapped around and still be perceived by the reader as typical of themselves. I’m not enormously familiar with the blood type hypothesis because it didn’t become established in Western culture to the same extent, and that fact that makes it possible to look at a possibly less rigorous model of personality from the outside. My issue with the Enneagram, and I’m not bothering you with the details here when maybe I should, is that it reminds me of the kind of nebulosity I associate with astrology, but it’s nevertheless reproducible and correlates to the MBTI. Why, then, are there two sets of personality theory, one inside academia and the other outside it? Another oddity I’ve noticed with this is that in spite of it being non-academic, MBTI tends to be popular with both sceptics and skeptics (see that link for an explanation regarding the distinction). That isn’t to say it isn’t valid, but it is odd.

Before I get back to agreeableness, I just want to mention personality disorders. Much of the general public is not keen on the concept of personality disorders, but I don’t want to address that right now because it’s quite involved. However, I do want to address the issue of specific personality disorder diagnoses, which are contentious for other reasons. One is that there are frequently several diagnoses possible within this cluster per client, and another is that people with the same diagnosis are often very dissimilar to each other. Therefore, another possible approach is dimensional, that is, to look at personality disorders in terms of extreme traits. There could, for example, be some common ground between the appallingly and basically uselessly named Borderline Personality Disorder and a high degree of neuroticism, although it doesn’t fit exactly, and other diagnosable personality disorders also correspond. It isn’t exact, but that may be a reflection of the inadequacy of the traditional model of personality disorders rather than a problem with this approach.

Back to agreeableness then.

The label “agreeableness” may be poorly named. Other options include likability, tendermindedness, friendly compliance and conformity. The last one really isn’t accurate, for me at least, as I am notably non-conformist, by which I mean I’m neither anti-conformist nor conformist – how many trans vegan evangelical Christians are there? I tend to feel that psychologists are really bad at naming things, for instance “theory of mind”, “schizophrenia” and “borderline personality disorder” are all abysmally named. Maybe “conformity” isn’t supposed to mean “conformism” here, I don’t know. But whatever you want to call it, it’s problematic. Leaving aside the obvious issue that agreeable people can suffer because of their niceness, which is hardly worth mentioning, there’s a separate problem with it because of how they interact with other people who are less agreeable. Bluntly, we tend to cave in to people who are not conscientious or open. This may even be bad for them. For instance, someone with disordered substance use such as alcoholism may find it easier to persuade an agreeable person to buy them alcohol or tobacco than they would if the person were less agreeable. The opposite of agreeableness could be described as antagonism, and sometimes antagonism is a good thing. For instance, antagonism could defeat an authoritarian society or a cult, and lead to greater assertiveness in relationships. In an abusive relationship, an agreeable person is more likely to suffer than one who is antagonistic. Agreeability can also lead to a kind of unholy alliance where a person’s malevolent influences are magnified. It could be, for example, that they can place an agreeable person at the front, interacting directly with people who can then be exploited, drawing others into a web of deceit, to use my own florid language. I expect you can all imagine your own scenarios, or perhaps it’s all too real.

Abstracting this from just this one personality trait, this is a good illustration of why it can make sense to be neutral about psychological phenomena. To some extent, most of us don’t understand child sexual abuse or paedophilia because we don’t want to understand it, and may feel that it would in some way defile us if we did. This leads to uninformed approaches to these two separate problems, confusion between the two and ineffective approaches to addressing them. Likewise, when I was younger I’d have been loath to admit that I was an agreeable person because it would’ve sounded like bragging. Now I’m aware of its possible negative aspects, it’s easier to own up, but that in itself is an emotive reaction – I’m owning up now because I can see its dark side. It would be better to try to be detached, when appropriate of course, from psychological concepts in order to be able to understand and use them better. Therefore, yes, I’m a highly agreeable person, for better or for worse, and like other personality traits it has its negative and positive aspects, as do other traits or positions on that dimension which may be seen as primarily negative.